Download Finding REMO - Detecting Relative Motion Patterns in Geospatial

Finding REMO - Detecting Relative Motion
Patterns in Geospatial Lifelines
Patrick Laube1, Marc van Kreveld2, Stephan Imfeld1
Department of Geography, University of Zurich, Winterthurerstrasse 190,
CH–8057 Zurich, Switzerland, {plaube,imfeld}@geo.unizh.ch1
Department of Computer Science, Utrecht University, P.O. Box 80.089,
3508 TB Utrecht, The Netherlands, [email protected]
Abstract
Technological advances in position aware devices increase the availability
of tracking data of everyday objects such as animals, vehicles, people or
football players. We propose a geographic data mining approach to detect
generic aggregation patterns such as flocking behaviour and convergence
in geospatial lifeline data. Our approach considers the object's motion
properties in an analytical space as well as spatial constraints of the object's lifelines in geographic space. We discuss the geometric properties of
the formalised patterns with respect to their efficient computation.
Keywords: Convergence, cluster detection, motion, moving point objects,
pattern matching, proximity
1 Introduction
Moving Point Objects (MPOs) are a frequent representation for a wide and
diverse range of phenomena: for example animals in habitat and migration
studies (e.g. Ganskopp 2001; Sibbald et al. 2001), vehicles in fleet management (e.g. Miller and Wu 2000), agents simulating people for modelling crowd behaviour (e.g. Batty et al. 2003) and even tracked soccer players on a football pitch (e.g. Iwase and Saito 2002). All those MPOs share
motions that can be represented as geospatial lifelines: a series of observations consisting of a triple of id, location and time (Hornsby and Egenhofer
2002).
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Patrick Laube, Marc van Kreveld, Stephan Imfeld
Gathering tracking data of individuals became much easier because of
substantial technological advances in position aware devices such as GPS
receivers, navigation systems and mobile phones. The increasing number
of such devices will lead to a wealth of data on space-time trajectories
documenting the space-time behaviour of animals, vehicles and people for
off-line analysis. These collections of geospatial lifelines present a rich environment to analyse individual behaviour. (Geographic) data mining may
detect patterns and rules to gather basic knowledge of dynamic processes
or to design location based services (LBS) to simplify individual mobility
(Mountain and Raper 2001; Smyth 2001; Miller 2003).
Knowledge discovery in databases (KDD) and data mining are responses to the huge data volumes in operational and scientific databases.
Where traditional analytical and query techniques fail, data mining attempts to distill data into information and KDD turns information into
knowledge about the monitored world. The central belief in KDD is that
information is hidden in very large databases in the form of interesting
patterns (Miller and Han 2001). This statement is true for the spatiotemporal analysis of geospatial lifelines and thus is a key motivator for the
presented research. Motion patterns help to answer the following type of
questions.
• Can we identify an alpha animal in the tracking data of GPS-collared
wolves?
• How can we quantify evidence of 'swarm intelligence' in gigabytes of
log-files from agent-based models?
• How can we identify which football team played the more catching lines
of defense in the lifelines of 22 players sampled at seconds?
The long tradition of data mining in the spatio-temporal domain is well
documented (for an overview see Roddick et al. (2001)). The Geographic
Information Science (GISc) community has recognized the potential of
Geographic Information Systems (GIS) to 'capture, represent, analyse and
explore spatio-temporal data, potentially leading to unexpected new
knowledge about interactions between people, technologies and urban infrastructures (Miller 2003). Unfortunately, most commercial GIS are based
on a static place-based perspective and are still notoriously weak in providing tools for handling the temporal dimensions of geographic information (Mark 2003). Miller postulates expanding GIS from the place-based
perspective to encompass a people-based perspective. He identifies the development of a formal representation theory for dynamic spatial objects
and of new spatio-temporal data mining and exploratory visualization
techniques as key research issues for GISc.
Finding REMO - Detecting Relative Motion Patterns in Geospatial Lifelines3
In this paper work is presented which extends a concept developed to
analyse relative motion patterns for groups of MPOs (Laube and Imfeld
2002) to also analyse the object's absolute locations. The work allows the
identification of generic formalised motion patterns in tracking data and
the extraction of instances of these formalised patterns. The significance of
these patterns is discussed.
2 Aggregation in Space and Time
Following Waldo Tobler's first law of geography, near things are more related than distant things (Tobler 1970). Tolber's law is often referred to as
being the core of spatial autocorrelation (Miller 2004). Nearness as a concept can be extended to include both space and time. Thus analysing geospatial lifelines we are interested in objects near in space-time. Objects that
are near at certain times might be related. Although correlation is not causality, it provides evidence of causality that can (and should) be assessed in
the light of theory and/or other evidence. Since this paper focuses on the
formal and geometrical definition and the algorithmic detection of motion
patterns we use geometric proximity in euclidian space to avoid the vague
term nearness.
To analyse geospatial lifelines this could mean that MPOs moving
within a certain range influence each other. E.g. an alpha wolf leads its
pack by being seen or heard, thus all wolves have to be located within the
range of vision or earshot respectively. Analysing geospatial lifelines we
are interested in first identifying motion patterns of individuals moving in
proximity. Second we want to know how, when and where sets of MPOs
aggregate, converge and build clusters respectively.
Investigating aggregation of point data in space and time is not new.
Most approaches focus on detecting localized clustering at certain time
slices (e.g. Openshaw 1994; Openshaw et al. 1999). This concept of spatial
clusters is static, rooted in the time sliced static map representation of the
world. With a true spatio-temporal view of the world aggregation must be
perceived as the momentary process convergence and the final static cluster as its possible result. The opposite of convergence, divergence, is
equally interesting. Its possible result, some form of dispersal, is much less
obvious and thus much harder to perceive and describe.
A cluster is not the compulsory outcome of a convergence process and
vice versa. A set of MPOs can very well be converging for a long time
without building a cluster. The 22 players of a football match may converge during an attack without ever forming a detectable cluster on the
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pitch. In reverse, MPOs moving around in a circle may build a wonderful
cluster but never be converging. In addition the process of convergence
and the final cluster are in many cases sequential. Consider the lifelines of
a swarm of bees. At sunset the bees move back to the hive from the surrounding meadows, showing a strong convergence pattern without building a spatial cluster. In the hive the bees wiggle around in a very dense
cluster, but do not converge anymore. In short, even though convergence
and clustering are often spatially and/or temporally tied up, there need not
be a detectable relation in an individual data frame under investigation.
3 The Basic REMO–Analysis Concept
The basic idea of the analysis concept is to compare the motion attributes
of point objects over space and time, and thus to relate one object's motion
to the motion of all others (Laube and Imfeld 2002). Suitable geospatial
lifeline data consist of a set of MPOs each featuring a list of fixes. The
REMO concept (RElative MOtion) is based on two key features: First, a
transformation of the lifeline data to a REMO matrix featuring motion attributes (i.e. speed, change of speed or motion azimuth). Second, matching
of formalized patterns on the matrix (Fig. 1).
Two simple examples illustrate the above definitions: Let the geospatial
lifelines in Fig. 1a be the tracks of four GPS-collared deer. The deer O1
moving with a constant motion azimuth of 45° during an interval t2 to t5,
i.e. four discrete time steps of length ∂t, is showing constance. In contrast,
four deer performing a motion azimuth of 45° at the same time t4 show
concurrence.
objects
time
t1 t2 t3 t4 t5
O1
O1
O2
O3
O4
time
t1 t2 t3 t4 t5
O1
O2
O3
O4
45 45 45
constance
90
45 45 45
45
90 315 45 315
0
45
45
90 180 45
45
90 315 90 45
0
45
45
45
45
concurrence
45 45 45
45
45
O3 O O
2 4
45
(a)
(b)
(c)
trend-setter
(d)
Fig. 1. The geospatial lifelines of four MPOs (a) are used to derive in regular intervals the motion azimuth (b). In the REMO analysis matrix (c) generic motion
patterns are matched (d).
Finding REMO - Detecting Relative Motion Patterns in Geospatial Lifelines5
The REMO concept allows construction of a wide variety of motion
patterns. See the following three basic examples:
• Constance: Sequence of equal motion attributes for r consecutive time
steps (e.g. deer O1 with motion azimuth 45° from t2 to t5).
• Concurrence: Incident of n MPOs showing the same motion attributes
value at time t (e.g. deer O 1, O2, O3, and O4 with motion azimuth 45° at
t4)
• Trend-setter: One trend-setting MPO anticipates the motion of n others.
Thus, a trend-setter consists of a constance linked to a concurrence (e.g.
deer O1 anticipates at t2 the motion azimuth 45° that is reproduced by all
other MPOs at time t4)
For simplicity we focus in the remainder of this paper on the motion attribute azimuth, even though most facets of the REMO concept are equally
valid for speed or change of speed.
4 Spatially Constrained REMO patterns
The construction of the REMO-matrix is an essential reduction of the information space. However it should be noted that this step factors out the
absolute locations of the fixes of the MPOs. The following two examples
illustrate generic motion patterns where the absolute locations must be
considered.
• Three geese all heading north-west at the same time – one over London,
one over Cardiff and one over Glasgow are unlikely to be influenced by
each other. In contrast, three geese moving north-west in the same gaggle are probably influenced. Thus, for flocking behaviours the spatial
proximity of the MPOs has to be considered.
• Three geese all heading for Leicester at the same time – one starting
over London, one over Cardiff and one over Glasgow show three different motion azimuths, not building any pattern in the REMO matrix.
Thus, convergence can only be detected considering the absolute locations of the MPOs.
The basic REMO concept must be extended to detect such spatially
constrained REMO patterns. In Section 4.1 spatial proximity is integrated
and in Section 4.2 an approach is presented to detect convergence in relative motion. Section 4.3 evaluates algorithmic issues of the proposed approaches.
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Patrick Laube, Marc van Kreveld, Stephan Imfeld
4.1 Relative Motion with Spatial Proximity
Many sheep moving in a similar way is not enough to define a flocking
pattern. We expect additionally that all the sheep of a flock graze on the
same hillside. Formalised as a generic motion pattern we expect for a
flocking the MPOs to be in spatial proximity. To test the proximity of m
MPOs building a pattern at a certain time we can compute the spatial
proximity of the m MPO's fixes in that time frame. Following Tobler's first
law, proximity among MPOs can be considered as impact ranges, or the
other way around: a spatio-temporally clustered set of MPOs is evidence to
suggest an interrelation among the involved MPOs.
The meaning of spatial constraint in a motion pattern is different if we
consider the geospatial lifeline of a single MPO. The consecutive observations (fixes) of a single sheep building a lifeline can be tested for proximity. Thus the proximity measure constrains the spatial extent of single object's motion pattern. A constance for a GPS-collared sheep may only be
meaningful if it spans a certain distance, excluding pseudo-motion caused
by inaccurate fix measurements.
Different geometrical and topological measures could be used to constrain motion patterns spatially. The REMO analysis concept focuses on
the following open list of geometric proximity measures.
• A first geometric constraint is the maximal length of the cumulated distances to the mean or median center (length of star plot).
• Another approach to indicate the spatial proximity of points uses the
Delaunay diagram, applied for cluster detection in 2-D point sets (e.g.
Estivill-Castro and Lee 2002) or for the visualisation of habitat-use intensity of animals (e.g. Casaer et al. 1999). According to the cluster detection approach two points belong to the same cluster, if they are connected by a small enough Delaunay edge. Thus, adapted to the REMO
concept a second distance proximity measure is to limit the average
length of the Delauney edges of a point group forming a REMO pattern.
• Proximity measures can have the form of bounding boxes, circles or ellipses (Fig. 2). The simplest way of indicating an impact range would be
to specify a maximal bounding box that enclosed all fixes relevant to the
pattern. Circular criteria can require enclosing all relevant fixes within
radius r or include the constraint to be spanned around the mean or median center of the fixes. Ellipses are used to rule the directional elongation of the point cluster (major axis a, minor axis b).
• Another areal proximity measure for a set of fixes is the indication of a
maximal border length of the convex hull.
Finding REMO - Detecting Relative Motion Patterns in Geospatial Lifelines7
Using these spatial constraints the list of basic motion patterns introduced in Section 3 can be amended with the spatially constrained REMO
patterns (Fig. 2).
• Track: Consists of the REMO pattern constance and the attachment of
spatial constraint. Definition: constance + spatial constraint S.
• Flock: Consists of the REMO pattern concurrence and the attachment of
a spatial constraint. Definition: concurrence + spatial constraint S.
• Leadership: Consists of the REMO pattern trend-setter and the attachment of a spatial constraint. For example the followers must lie within
the range (∂x, ∂y) when they join the motion of the trend-setter. Definition: trend-setter + spatial constraint S.
ANALYSIS SPACE
(a)
time
∂ymax
(b)
rmax
objects
track
GEOGRAPHIC SPACE
∂xmax
t3
t2
t1
to
objects
flock
time
r
a
∂y
b
∂x
objects
t3
to
t1
t2
∂xmax
leadership
time
∂ymax
Fig. 2. The figure illustrates the constraints of the patterns track, flock and
leadership in the analysis space (the REMO matrix) and in the geographic space.
Fixes matched in the analysis space are represented as solid forms, fixes not
matched as empty forms. Some possible spatial constraints are represented as
ranges with dashed lines. Whereas in the situations (a) the spatial constraints for
the absolute positions of the fixes are fulfilled they are not in the situations (b):
For track the last fix lies beyond the range, for flock and leadership the quadratic
object lies outside the range.
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4.2 Convergence
At the same time self-evident and fascinating are groups of MPOs aggregating and disaggregating in space and time. An example is wild animals
suddenly heading in a synchronised fashion for a mating place. Wildlife
biologists could be interested in the who, when and where of this motion
pattern. Who is joining this spatio-temporal trend? Who is not? When does
the process start, when does it end? Where lies the mating place, what spatial extent or form does it have? A second example comes from the analysis of crowd behaviour. Can we identify points of interest attracting people
only at certain times, events of interest rather than points of interest, losing
their attractiveness after a while? To answer such questions we propose the
spatial REMO pattern convergence.
The phenomenon aggregation has a spatial and a spatio-temporal form.
An example may help to illustrate the difference. Let A be a set of n antelopes. A wildlife biologist may be interested in identifying sets of antelopes heading for some location at certain time. The time would indicate
the beginning of the mating season, the selected set of m MPOs the readyto-mate individuals, and the spot might be the mating area. This is a convergence pattern. It is primarily spatial, that means the MPOs head for an
area but may reach it at different times. On the other hand the wildlife biologist and the antelopes may share the vital interest to identify MPOs that
head for some location and actually meet there at some time extrapolating
their current motion. Thus, the pattern encounter includes considerations
about speed, excluding MPOs heading for a meeting range but not arriving
there at a particular time with the others.
• Convergence: Heading for R. Set of m MPOs at interval i with motion
azimuth vectors intersecting within a range R of radius r.
• Encounter: Extrapolated meeting within R. Set of m MPOs at interval i
with motion azimuth vectors intersecting within a range R of radius r
and actually meeting within R extrapolating the current motion.
Finding REMO - Detecting Relative Motion Patterns in Geospatial Lifelines9
r
t0
t1
1
t2
t3
t4
t6
2
2
3
t5
4
3
2
r
3
2
Fig. 3. Geometric detection of convergence. Let S be a set of 4 MPOs with 7 fixes
from t0 to t6. The illustration shows a convergence pattern found with the parameters 4 MPOs at the temporal interval t1 to t3. The darkest polygon denotes an area
where all 4 direction vectors are passing at a distance closer than r. The pattern
convergence is found if such a polygon exists. Please note that the MPOs do not
build a cluster but nevertheless show a convergence pattern.
The convergence pattern is illustrated in Figure 3. Let S be a set of
MPOs with n fixes from t0 to t n-1. For every MPO and for every interval of
length i an azimuth vector fitting in its fixes within i represents the current
motion. The azimuth vector can be seen as a half-line projected in the direction of motion. The convergence is matched if there is at any time a circle of radius r that intersects n directed half-lines fitted for each MPO in
the fixes within i. For the encounter pattern whether the objects actually
meet in future must additionally be tested.
The opposites of the above described patterns are termed divergence and
breakup. The latter term integrates a spatial divergence pattern with the
temporal constraint of a precedent meeting in a range R. The graphical representation of the divergence pattern is highly similar to Fig. 3. The only
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difference lies in the construction of the strips, heading backwards instead
of forwards, relative to the direction of motion.
4.3 Algorithms and Implementation Issues
In this section we develop algorithms to detect the above introduced motion patterns and analyse their efficiency.
The basic motion patterns in the REMO concept are relatively easy to
determine in linear time. The addition of positions requires more complex
techniques to obtain efficient algorithms. We analyse the efficiency of
pattern discovery for track, flock, leadership, convergence, and encounter
in this section. We let the range be a circle of given radius R. Let n denote
the number of MPOs in the data set, and t the number of time steps. The
total input size is proportional to nt, so a linear time algorithm requires
O(nt) time. We let m denote the number of MPOs that must be involved in
a pattern to make it interesting. Finally, we assume that the length of a
time interval is fixed and given.
The addition of geographic position to the REMO framework requires
the addition of geographic tests or the use of geometric algorithms. The
track pattern can simply be tested by checking each basic constance pattern found for each MPO. If the constance pattern also satisfies the range
condition, a track pattern is found. The test takes constant additional time
per pattern, and hence the detection of track patterns takes O (nt) time
overall.
Efficient detection of the flock pattern is more challenging. We first
separate the input data by equal time and equal motion direction, so that
we get a set of n' ≤ n points with the same direction and at the same time.
The 8t point sets in which patterns are sought have total size O(nt). To discover whether a subset of size at least m of the n points lie close together,
within a circle of radius R, we use higher-order Voronoi diagrams. The mth order Voronoi diagram is the subdivision of the plane into cells, such
that for any point inside a cell, some subset of m points are the closest
among all the points. The number of cells is O(m(n'-m)) (Aurenhammer
1991), and the smallest enclosing circle of each subset of m points can be
determined in O(m) time (de Berg et al. 2000, Sect. 4.7). If the smallest
enclosing circle has radius at most R, we have discovered a pattern. The
sum of the n' values over all 8t point sets is O(nt), so the total time needed
to detect these patterns is O(ntm2 + nt log n). This includes the time to
compute the m-th order Voronoi diagram (Ramos 1999).
Leadership pattern detection can be seen as an extension of flock pattern
detection. The additional condition is that one of the MPOs shows con-
Finding REMO - Detecting Relative Motion Patterns in Geospatial Lifelines11
stance over the previous time steps. Leadership detection also requires
O(ntm2+nt log n) time.
For the convergence pattern, consider a particular time interval. The n
MPOs give rise to n azimuth vectors, which we can see as directed halflines. To test whether at least m MPOs out of n converge, we compute the
arrangement formed by the thickened half-lines, which are half-strips of
width 2r. For every cell in the arrangement we determine how many thickened half-lines contribute, which can be done by traversing the arrangement once and maintaining a counter that shows in how many half-strips
the current cell is. If a cell is contained in at least m half-strips, it constitutes a pattern. Computing the arrangement of n half-strips and setting the
counters can be done in O(n2) time in total; the algorithm is very similar to
computing levels in arrangements (de Berg et al. 2000, Chap. 8). Since we
consider t different time intervals, the total running time becomes O(n2t).
The encounter pattern is the most complex one to compute. The reason
is that extrapolated meeting times must also match, which adds a dimension to the space in which geometric algorithms are needed. We lift the
problem into 3-D space, where the third dimension is time. The MPOs become half-lines that go upward from a common horizontal plane representing the beginning of the time interval; the slope of the half-lines will
now be the speed. The geometric problem to be solved is finding horizontal circles of radius R that are crossed by at least m half-lines, which can be
solved in O(n4) time with a simple algorithm. For all time intervals of a
given length, the algorithm needs O(n4t) time.
5 Discussion
The REMO approach has been designed to analyse motion basing on geospatial lifelines. Since motion is expressed by a change in location over
time the REMO patterns intrinsically span over space and time. Our approach thus overcomes the limitation of only either detecting spatial clusters on snapshots or highlighting temporal trends in attributes of spatial
units. It allows pattern detection in space-time.
REMO patterns rely solely on point observations and are thus expressible for any objects that can be represented as points and leave a track in a
euclidean space. Having translated the expected behaviours into REMO
patterns, the detection process runs unsupervised, listing every pattern occurrence. The introduced patterns can be detected within reasonable time.
Many simple patterns can be detected in close to linear time if the size of
the subset m that constitutes a pattern is a constant, which is natural in
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Patrick Laube, Marc van Kreveld, Stephan Imfeld
many situations. The encounter pattern is quite expensive to compute, but
since we focus on off-line analysis, we can still deal with data sets consisting of several hundreds of MPOs. Note that the dependency on the
number of time steps is always linear for fixed length time intervals. The
most promising way to obtain more efficient algorithms is by using approximation algorithms, which can save orders of magnitude by stating the
computational problem slightly less firm (Bern and Eppstein 1997). In
short, the REMO concept can cope with the emerging data volumes of
tracking data.
Syntactic pattern recognition adopts a hierarchical perspective where
complex patterns are viewed as being composed of simple primitives and
grammatical rules (Jain et al. 2000). Sections 3 and 4 introduced a subset
of possible pattern primitives of the REMO analysis concept. Using the
primitives and a pattern description formalism almost arbitrary motion
patterns can be described and detected. Due to this hierarchical design the
concept easily adapts to the special requirements of various application
fields. Thus, the approach is flexible and universal, suited for various lifelines such as of animals, vehicles, people, agents or even soccer players.
The detection of patterns of higher complexity requires more sophisticated
and flexible pattern matching algorithms than the ones available today.
The potential users of the REMO method know the phenomenon they
investigate and the data describing it. Hence, in contrast to traditional data
mining assuming no prior knowledge, the users come up with expectations
about conceivable motion patterns and are able to assess the results of the
pattern matching process. Therein lies a downside of the REMO pattern
detection approach: It requires relatively sophisticated knowledge about
the patterns to be searched for. For instance, the setting of an appropriate
impact range for a flock pattern is highly dependent on the investigated
process and thus dependent on the user. In general the parametrisation of
the spatial constraints influences the number of patterns detected. Further
research is needed to see whether autocalibration of pattern detection will
be possible within the REMO concept.
Even though the REMO analysis concept assumes users specifying patterns they are interested in, the pattern extent can also be viewed as an
analysis parameter of the data mining approach. One reason to do so is to
detect scale effects lurking in different granularities of geospatial lifeline
data. The number of matched patterns may be highly dependent on the
spatial, temporal and attributal granularity of the pattern matching process.
For example the classification of motion azimuth in only the two classes
east and west reveals a lot of presumably meaningless constance patterns.
In contrast, the probability of finding constance patterns with 360 azimuth
Finding REMO - Detecting Relative Motion Patterns in Geospatial Lifelines13
classes is much smaller, or take the selection of the impact range r for the
flock pattern in sheep as another example. By testing the length of the impact range r against the amount of matched patterns one could search for a
critical maximal impact range within a flock of sheep. Future research will
address numerical experiments with various data to investigate such relations.
A critical issue in detecting convergence is fitting the direction vector in
a set of fixes. Only slight changes in its azimuth may have huge effects on
the overlapping regions. A straightforward solution approach to this problem is to smooth the lifelines and then fitting the azimuth vector to a segment of the smoothed lifeline.
The paper illustrates the REMO concept referring to ideal geospatial
lifeline data. In reality lifeline data are often imprecise and uncertain. Sudden gaps in lifelines, irregular fixing intervals or positional uncertainty of
fixes requires sophisticated interpolation and uncertainty considerations on
the implementation side (e.g. Pfoser and Jensen 1999).
6 Conclusions
With the technology driven shift from the static map view of the world to a
dynamic process in GIScience, cluster detection on snapshots is insufficient. What we need are new methods that can detect convergence processes as well as static clusters, especially if these two aspects of space-time
aggregation are separated. We propose a generic, understandable and extendable approach for data mining in geospatial lifelines. Our approach
integrates individuals as well as groups of MPOs. It also integrates parameters describing the motion as well as the footprints of the MPOs in
space-time.
Acknowledgements
The ideas to this work prospered in the creative ambience of Dagstuhl
Seminar No. 03401 on 'Computational Cartography and Spatial Modelling'. The authors would like to acknowledge invaluable input from Ross
Purves, University of Zurich.
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Patrick Laube, Marc van Kreveld, Stephan Imfeld
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